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Let $X$ be a variety and $\mathcal{L}$ be a very ample line bundle on $X$. Suppose $H^0(X,\mathcal{L}) = \langle s_0,...,s_n \rangle$ then there is an immersion into projective space:

$$ X \rightarrow \mathbb{P}(\,H^0(X,\mathcal{L})\check \,)$$

given by evaluation on closed points $x\rightarrow \{s\in H^0(X,\mathcal{L})\,|\,s(x)=0\}$.

My first question is this: how can I extend the definition of this morphism to non-closed points? I know there is alternative approach as in Hartshorne, however I was looking for a choice free method to define such morphism.

My second question is somewhat related: it is often said that $\mathcal{L}$ gives rise to a morphism

$$ X \longrightarrow \mathrm{Proj}\left(\oplus_{k\in\mathbb{N}} \,H^0(X,\mathcal{L}^{\otimes k}\,)\right)\longrightarrow \mathbb{P}_k^n = \mathrm{Proj}\left(\,k\,[s_0,s_1,...,s_n]\,\right) $$

where I believe the first map is an isomorphism (but I am not sure) and the second map is an immersion into projective space (again I am not sure you would want to define projective space in this way). How can you see this fact? I cannot find any references so feel welcome to just tell me where to look it up.

Thank you!

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  • $\begingroup$ For first question, can't you just extend this to a nonclosed point with the obvious: if $\eta$ is a nonclosed point, define its image by closing off image(closed point in $\bar\eta$)? $\endgroup$ Nov 14, 2018 at 14:46

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May be this will help. Let $X$ be a variety over a field $k$, $L$ a line bundle on $X$, $V$ a finite dimensional vector space over $k$ and assume you have a surjection $V\otimes_k\mathcal{O}_X\to L$. This gives by universal property of projective spaces, a morphism $\phi:X\to \mathbb{P}(V^*)=P$, the natural surjection $V\otimes_k\mathcal{O}_P\to\mathcal{O}_P(1)$ pulls back to the surjection you started with. In your situation, take $V=H^0(X,L)$. The definition does not make use of closed or non-closed points.

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